We prove new one shot achievability results for measurement compression of quantum instruments with side information at the receiver. Unlike previous one shot results for this problem, our one shot bounds are nearly optimal and do not need catalytic randomness. In fact, we state a more general problem called centralised multi link measurement compression with quantum side information and provide one shot achievability results for it. As a simple corollary, we obtain one shot measurement compression results for quantum instruments with side information that we mentioned earlier. All our one shot results lead to the standard results for this problem in the asymptotic iid setting. We prove our achievability bounds by first proving a novel sequential classical quantum multipartite covering lemma, which should be of independent interest.

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